On a crystalline variational problem, part I: first variation and global $L^\infty$ regularity

Let $\phi : \ensuremath{\mathbb R^n} \to [0,+\infty[$ be a given
positively one-homogeneous convex function, and let
$\mathcal{W}_\phi := \{\phi \leq 1\}$. Motivated by motion by crystalline
mean curvature in three dimensions, we introduce and study the
class ${\mathcal R}_\phi(\ensuremath{\mathbb R^n} )$ of ``smooth''
boundaries in the relative geometry induced by the ambient Banach
space $(\ensuremath{\mathbb R^n} , \phi)$. One can realize that,
even when $\mathcal{W}_\phi$ is a polytope,
${\mathcal R}_\phi({\ensuremath{\mathbb R^n}})$ cannot be reduced
to the class of polyhedral
boundaries (locally resembling $\partial \mathcal{W}_\phi$). Curved
portions must be necessarily included and this fact (as well as the
nonsmoothness of $\partial \mathcal{W}_\phi$) is source of several
technical difficulties, related to the geometry of Lipschitz manifolds.
Given a boundary $\partial E$ in the class ${\mathcal
R}_\phi({\ensuremath{\mathbb R^n}})$, we rigorously compute the
first variation of the corresponding anisotropic perimet which leads to
a variational problem on vector fields defined on $\partial E$. It turns
out that the minimizers have a uniquely determined
(intrinsic) tangential divergence on $\partial E$. We define such a
divergence to be the $\phi$-mean curvature $\kappa_\phi$ of $\partial
E$; the function $\kappa_\phi$ is expected to be the initial velocity of
$\partial E$, whenever $\partial E$ is considered as the initial datum
for the corresponding anisotropic mean curvature flow.
We prove that $\kappa_\phi$ is bounded on $\partial E$ and that
its sublevel sets are characterized through a variational inequality.